Simplify the Matrix

Problem

[[3,2,−1,−15],[5,3,2,0],[3,2,3,11],[−6,−4,2,40]]

Solution

1. Identify the goal to simplify the matrix into row-echelon form using elementary row operations.

2. Eliminate the entries in the first column below the first row by performing (R_2)→(R_2)−5/3*(R_1) (R_3)→(R_3)−(R_1) and (R_4)→(R_4)+2*(R_1)

[[3,2,−1,−15],[0,−1/3,11/3,25],[0,0,4,26],[0,0,0,10]]

3. Normalize the first row by dividing by 3 and the second row by multiplying by −3

[[1,2/3,−1/3,−5],[0,1,−11,−75],[0,0,4,26],[0,0,0,10]]

4. Simplify the third and fourth rows by dividing by their leading coefficients.

[[1,2/3,−1/3,−5],[0,1,−11,−75],[0,0,1,6.5],[0,0,0,1]]

5. Perform back-substitution to reach reduced row-echelon form, starting with the fourth column.

[[1,2/3,−1/3,0],[0,1,−11,0],[0,0,1,0],[0,0,0,1]]

6. Continue back-substitution for the third and second columns.

[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]

Final Answer

[[3,2,−1,−15],[5,3,2,0],[3,2,3,11],[−6,−4,2,40]]=[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]

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